Extreme values for Sn(σ,t) near the critical line

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EXTREME VALUES FOR S_n(σ, t) NEAR THE CRITICAL LINE

ANDR ´ES CHIRRE

Abstract. LetS(σ, t) = ¹_πargζ(σ+it) be the argument of the Riemann zeta function at the pointσ+it of the critical strip. Forn≥1 andt >0 we define

Sn(σ, t) = Z t

0

Sn−1(σ, τ) dτ +δn,σ,

whereδn,σ is a specific constant depending onσandn. Let 0≤β <1 be a fixed real number. Assuming the Riemann hypothesis, we show lower bounds for the maximum of the functionSn(σ, t) on the interval T^β≤t≤T and near to the critical line, whenn≡1 mod 4. Similar estimates are obtained for|Sn(σ, t)|

whenn6≡1 mod 4. This extends the results of Bondarenko and Seip [7] for a region near the critical line.

In particular we obtain some omega results for these functions on the critical line.

1. Introduction

In this paper, following similar ideas from Bondarenko and Seip [7], we obtain new estimates for extreme values of the argument of the Riemann zeta function and its antiderivatives near the critical line assuming the Riemann hypothesis. Our main tools are convolution formulas for the functionsS_n(σ, t) and the version of the resonance method of Bondarenko and Seip given in [7].

Let us begin by defining the main objects of our study and some results of them.

1.1. Background. Letζ(s) denote the Riemann zeta function. For ¹₂ ≤σ≤1 andt >0 we define S(σ, t) = ¹_πargζ σ+it

,

where the argument is obtained by a continuous variation along straight line segments joining the points 2, 2 +itand σ+it, assuming that this path has no zeros ofζ, with the convention that argζ(2) = 0. If this path has zeros ofζ (including the endpointσ+it) we set

S(σ, t) =¹₂ lim

ε→0{S(σ, t+ε) +S(σ, t−ε)}.

Useful information on the qualitative and quantitative behavior of S(σ, t) is encoded in its antiderivatives.

SettingS₀(σ, t) :=S(σ, t), forn≥1 we define, inductively, the functions S_n(σ, t) =

Z t

0

S_n−1(σ, τ) dτ +δ_n,σ, whereδn,σ is a specific constant depending onσandn. These are given by

δ_2k−1,σ =(−1)^k−1 π

Z ∞

σ

Z ∞

u2k−1

. . . Z ∞

u3

Z ∞

u2

log|ζ(u1)|du1du2. . . du_2k−1

2010Mathematics Subject Classification. 11M06, 11M26, 11N37.

Key words and phrases. Riemann zeta function, Riemann hypothesis, argument.

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forn= 2k−1, withk≥1, and δ_2k,σ= (−1)^k−1

Z 1

σ

Z 1

u_2k

. . . Z 1

u3

Z 1

u2

du₁du₂ . . .du_2k= (−1)^k−1(1−σ)^2k (2k)!

forn= 2k, with k≥1.

Letn≥0 be an integer and ¹₂ ≤σ≤1 be a fixed real number. We extend the functions t7→Sn(σ, t) to Rin such a way thatSn(σ, t) is an odd function whennis even or is an even function whennis odd.

1.2. Behavior on the critical line. When σ = ¹₂, we use the classical notation Sn(¹₂, t) = Sn(t) and S₀(t) =S(t). In 1924, J. E. Littlewood [16, Theorem 11] established, under the Riemann hypothesis (RH), the bound¹

Sn(t) =On

logt (log logt)ⁿ⁺¹

, (1.1)

forn≥0. The order of magnitude of (1.1) has never been improved and the efforts have been concentrated on optimizing the value of the implicit constants. The best known result for n = 0 and n = 1 is due to Carneiro, Chandee and Milinovich [10] and forn≥2 is due to Carneiro and Chirre [11].

On the other hand, forn= 0 we have the following omega results² S(t) = Ω_± (logt)¹²

(log logt)¹²

!

, (1.2)

established by Montgomery [19, Theorem 2], under RH. It is likely that the estimate (1.2) is closer to the behavior of the functionS(t) than the estimate (1.1). In fact, a heuristic argument by Farmer, Gonek and Hughes [13] suggests thatS(t) grows as (logtlog logt)¹². Similarly, for the functionS1(t) Tsang [23, Theorem 5] established, under RH, that

S1(t) = Ω_± (logt)¹² (log logt)³²

! .

For the casen≥2, there are no known omega results forS_n(t).

Recently, Bondarenko and Seip used their version of the resonance method with a certain convolution formula forζ(s) to produce large values of the Riemann zeta function on the critical line [7]. Besides, using a convolution formula for logζ(s), they obtained similar results for the functionsS(t) andS1(t). They showed the following theorem.

Theorem 1 (cf. Bondarenko and Seip [7]). Assume the Riemann hypothesis. Let 0≤β <1 be a fixed real number. Then there exist two positive constants c0 andc1 such that, wheneverT is large enough,

max

T^β≤t≤T|S(t)| ≥c₀(logT)¹²(log log logT)¹² (log logT)¹² and

max

T^β≤t≤TS₁(t)≥c₁(logT)¹²(log log logT)¹² (log logT)³² .

1The notationf=O(g) (orfg) means|f(t)| ≤C g(t) for some constantC >0 andtsufficiently large. In the subscript we indicate the parameters in which such constantCmay depend on.

2The notationf= Ω+(g) meansf(t)> C g(t) for some constantC >0 and for some arbitrarily large values oft. The notation f= Ω−(g) meansf(t)<−C g(t) for some constantC >0 and for some arbitrarily large values oft. The notationf = Ω±(g) means thatf= Ω+(g) andf= Ω−(g).

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Theorem 1 implies the following omega result³forS(t):

S(t) = Ω (logt)¹²(log log logt)¹² (log logt)¹²

! .

This result can be compared with the Ω_± results of Montgomery. ForS1(t), Theorem 1 improved the Ω+

result given by Tsang by a factor (log log logt)¹².

1.3. Behavior in the critical strip. Recently Carneiro, Chirre and Milinovich [12, Theorem 2] showed new estimates for S_n(σ, t) similar to (1.1). In particular, for a fixed number ¹₂ < σ <1, under RH, we have that

Sn(σ, t) =On,σ

(logt)^2−2σ (log logt)ⁿ⁺¹

,

forn≥0. On the other hand, under RH, Tsang [23, Theorem 2 and p. 382] states the following lower bound sup

t∈[T ,2T]

±S(σ, t)≥c (logT)¹²

(log logT)¹², (1.3)

for ¹₂ ≤σ ≤ ¹₂+ _{log log}¹ _T, T sufficiently large and some constantc > 0. This result shows extreme values for S(σ, t) near the critical line. For the critical strip, a result of Montgomery [19] states that, for a fixed

1

2 < σ <1, we have

S(σ, t) = Ω_±

(σ−¹₂)² (logt)^1−σ (log logt)^σ

.

The main result of this paper is to show lower bounds forSn(σ, t) near the critical line, similar to (1.3).

Theorem 2. Assume the Riemann hypothesis. Let0≤β <1be a fixed number. Letσ >0be a real number andT >0 sufficiently large in the range

1

2 ≤σ≤ 1

2 + 1

log logT.

Then there exists a sequence {cn}n≥0 of positive real numbers with the following property.

(1) Ifn≡1 mod 4:

max

T^β≤t≤TSn(σ, t)≥cn

(logT)^1−σ(log log logT)^σ (log logT)^σ+n . (2) In the other cases:

max

T^β≤t≤T|Sn(σ, t)| ≥cn

(logT)^1−σ(log log logT)^σ (log logT)^σ+n .

Note that whenσ=¹₂ andn= 0 or 1, we recover Theorem 1. Moreover, we obtain the new omega results on the critical line.

Corollary 3. Assume the Riemann hypothesis. Then (1) Ifn≡1 mod 4:

S_n(t) = Ω₊ (logtlog log logt)¹² (log logt)ⁿ⁺¹²

! .

3The notationf= Ω(g) means that limt→∞f(t)/g(t)6= 0.

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(2) In the other cases:

Sn(t) = Ω (logtlog log logt)¹² (log logt)ⁿ⁺¹²

! .

1.4. Strategy outline. Our approach is motivated by the ideas of Bondarenko and Seip [7] on the use of their version of the resonance method and a convolution formula for logζ(s). Soundararajan [21] introduced the resonance method to produce large values of the Riemann zeta function on the critical line and large and small central values ofL-functions. Also, this method has been the main tool for finding large values for the Riemann zeta function,L-functions and other objects related to them, in the critical strip (for instance in [1, 2, 3, 4, 5, 6, 7, 8, 9, 15, 18]). The main idea of the resonance method is to find a certain Dirichlet polynomial which “resonates” with the object to study. The use of this Dirichlet polynomial is the principal difference between [7] and the works of Selberg and Tsang, where they used estimates of high moments to detect large values of Dirichlet series. Also, in contrast to the resonance method of Soundararajan [21], Bondarenko and Seip used significantly larger primes and a longer Dirichlet polynomial in the resonator. We will construct this Dirichlet polynomial in Section 4.

The strategy can be broadly divided into the following three main steps:

1.4.1. Step 1: Some results forSn(σ, t). The first step is to show bounds forSn(σ, t) and for their moments.

Bondarenko and Seip only needed to use the Littlewood’s estimate (1.1) and bounds of Selberg [22] for the moments ofS(t) andS1(t), assuming the Riemann hypothesis. In our case, we will use a weaker version of the result of Carneiro, Chirre and Milinovich [12], to estimate the functionSn(σ, t) uniformly in the critical strip. As a simple consequence of this result, we will obtain an estimate for its first moment. Finally, we will extend the convolution formula for logζ(s) given in [23, Lemma 5] for the functionSn(σ, t). Although we restrict our attention to a region close to the critical line, we will show the bounds forS_n(σ, t) in the critical strip, which may be of interest for other applications.

1.4.2. Step 2: The resonator. The construction of our resonator is similar to that made by Bondarenko and Seip [7, Section 3]. In particular, when σ= ¹₂ we obtain the resonator used by them. A deeper analysis in [7, Lemmas 3 and 4] allows us to show these results for a region close to the critical line. This implies that the main relation between the resonator and the convolution formula of Sn(σ, t) will follow immediately in the same way as obtained in the caseσ= ¹₂ [7, Lemma 7].

1.4.3. Step 3: Proof of Theorem 2. We follow the same outline in the proof of [7, Theorem 2]. We will estimate the error terms in the integral that contains the resonator and the convolution formula ofSn(σ, t).

The main difference in our proof with that of Bondarenko and Seip is in the choice of the sign for a certain Gaussian kernel. This choice will depend on the remainder ofnmodulo 4. In particular, this allows to obtain Ω₊ results forS_n(t) whenn≡1 mod 4 and Ω results in the other cases.

Throughout this paper we will assume the Riemann hypothesis. Besides, for f ∈ L¹(R), we define the Fourier transformfbby

fb(ξ) = Z ∞

−∞

f(x)e^−2πiξxdx, forξ∈R.

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2. Some results for Sn(σ, t)

The main goal in this section is to show bounds for the functionsSn(σ, t) and some convolution formulas of these functions with certain kernels. Throughout this section we let n≥0 be an integer and 0< δ ≤ ¹₂ be a real number.

2.1. Bounds for Sn(σ, t). The bounds that we will use for the functions Sn(σ, t) will be a weaker version of a result of Carneiro, Chirre and Milinovich [12].

Theorem 4. Assume the Riemann hypothesis. We have the uniform bound Sn(σ, t) =On,δ

(logt)^2−2σ (log logt)ⁿ⁺¹

in ¹₂ ≤σ≤1−δ <1and t >0 sufficiently large. In particular, we obtain for allt∈Rthat

S_n(σ, t) =O_n,δ(log(|t|+ 2)). (2.1)

Proof. It is enough to show whenσ > ¹₂. Fortsufficiently large we have that (1−σ)²log logt≥δ²log logt≥1.

Then, by [12, Theorem 2] we have

−C_n,σ⁻ (t) +On,δ(1)(log logt)^2−2σ

(log logt)ⁿ⁺¹ ≤Sn(σ, t)≤

C_n,σ⁺ (t) +On,δ(1)(log logt)^2−2σ

(log logt)ⁿ⁺¹ , (2.2) whereC_n,σ^± (t) are positive functions. Forn≥1 odd, these functions are given by:

C_n,σ^± (t) = 1 2ⁿ⁺¹π

Hn+1

±(−1)ⁿ⁺¹² (logt)^1−2σ

+ 2σ−1 σ(1−σ)

, (2.3)

where

Hn(x) = 1 +

∞

X

k=1

x^k (k+ 1)ⁿ. Note that whenm≥2, we have the bounds

1− 1

2^m ≤Hm(x)≤ζ(m),

for|x| ≤1. Therefore, we obtain in (2.3) forn≥1 odd andt sufficiently large

an,δ ≤C_n,σ^± (t)≤bn,δ, (2.4)

for some positive constants a_n,δ and b_n,δ. Using (2.2) we obtain the desired result in this case. Forn≥2 even, these functionsC_n,σ^± (t) are given by:

C_n,σ^± (t) = 2 C_n+1,σ⁺ (t) +C_n+1,σ⁻ (t)

C_n−1,σ⁺ (t)C_n−1,σ⁻ (t) C_n−1,σ⁺ (t) +C_n−1,σ⁻ (t)

!¹₂ .

Since (2.4) holds forC_n−1,σ^± (t) andC_n+1,σ^± (t), we have a similar estimate for C_n,σ^± (t), and this implies the desired result in this case. Whenn= 0 we have that

C_0,σ^± (t) =

2 C_1,σ⁺ (t) +C_1,σ⁻ (t)

C_−1,σ(t)¹₂ ,

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where the functionC−1,σ(t) is defined by C_−1,σ(t) = 1

π

1

1 + (logt)^1−2σ + 2σ−1 σ(1−σ)

.

Using (2.4) and a simple bound forC_−1,σ(t), we boundC_0,σ^± (t) and we conclude. Thefefore, it follows easily that (2.1) is valid for t≥t0 where t0 is sufficiently large, and using the fact that the functions Sn(σ, t) are

bounded in [¹₂,1−δ]×[0, t0] we conclude the proof.

As a simple consequence we have the following estimate Z T

0

|S_n(σ, t)|dt=O_n,δ(TlogT), (2.5)

uniformly in ¹₂ ≤σ≤1−δ <1 andT ≥2. Although this estimate is weak, it is sufficient for our purposes.

For the caseσ= ¹₂, better estimates are given by Littlewood [17, Theorem 9 and p. 179] for alln≥0.

2.2. Convolution formula. Now, we will obtain convolution formulas for the functionsSn(σ, t) with certain kernels. The next lemma was introduced by Selberg [22], and was also used by Tsang to study the functions S(t) andS₁(t) [23, 24]. Since we assume the Riemann hypothesis, the factor that contains the zeros outside the critical line disappears.

Lemma 5. Assume the Riemann hypothesis. Suppose that ¹₂ ≤σ ≤ 2, and let K(x+iy) be an analytic function in the horizontal stripσ−2≤y≤0satisfying the growth estimate

Vσ(x) := max

σ−2≤y≤0|K(x+iy)|=O 1

|x|log²|x|

when|x| → ∞. Then for every t6= 0, we have Z ∞

−∞

logζ(σ+i(t+u))K(u)du=

∞

X

m=2

Λ(m) m^σ+itlogmKb

logm 2π

+O Vσ(−t)

. (2.6)

Proof. See [23, Lemma 5].

It is clear that the above lemma gives a convolution formula for the functionS(σ, t). To obtain a similar formula for the functionSn(σ, t) whenn≥1, we need an expression that connects the functionSn(σ, t) with logζ(s).

Lemma 6. For ¹₂ ≤σ≤1 andt6= 0we have S_n(σ, t) = 1

π Im iⁿ

(n−1)!

Z ∞

σ

(u−σ)ⁿ⁻¹ logζ(u+it) du

.

Proof. This follows from [12, Lemma 6] and integration by parts.

Using this expression we obtain the following convolution formula. This generalizes Tsang’s conditional formula in [24] (or [7, Eq. (10)].

Proposition 7. Assume the Riemann hypothesis and the same conditions for the functionK(x+iy)as in Lemma 5. Suppose further that K is an even real-valued function (or odd real-valued function). Then for

1

2 ≤σ≤1 andt6= 0, we have Z ∞

−∞

Sn(σ, t+s)K(s)ds= 1 πIm

iⁿ

∞

X

m=2

Λ(m)

m^σ+it(logm)ⁿ⁺¹Kb logm

2π

+On V1

2(t) +||K||1

.

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Proof. For the casen= 0, we only need to take imaginary parts in (2.6). Forn≥1, by Lemma 6 we get Sn(σ, t) = 1

π Im iⁿ

(n−1)!

Z 2

σ

(u−σ)ⁿ⁻¹logζ(u+it) du

+On(1).

Plugging this in Lemma 5 we obtain Z ∞

−∞

S_n(σ,t+s)K(s)ds

= 1 π

Z ∞

−∞

Im iⁿ

(n−1)!

Z 2

σ

(u−σ)ⁿ⁻¹ logζ(u+i(t+s)) du

K(s)ds+On ||K||1

= 1 πIm

( iⁿ (n−1)!

Z 2

σ

(u−σ)ⁿ⁻¹ Z ∞

−∞

logζ(u+i(t+s))K(s)ds

! du

)

+On ||K||1

= 1 πIm

( iⁿ (n−1)!

Z 2

σ

(u−σ)ⁿ⁻¹

∞

X

m=2

Λ(m) m^u+itlogmKb

logm 2π

! du

)

+On V1

2(t) +||K||1

= 1 πIm

( iⁿ (n−1)!

∞

X

m=2

Λ(m) m^itlogmKb

logm 2π

Z 2

σ

(u−σ)ⁿ⁻¹ m^u du

)

+O_n V1

2(t) +||K||₁ ,

(2.7)

where the interchange of the integrals is justified by Fubini’s theorem, considering the estimates [20, Theorem 13.18, Theorem 13.21]. Using [14,§2.321 Eq.2]) we obtain that

Z 2

σ

(u−σ)ⁿ⁻¹

m^u du= βn−1

m^σ(logm)ⁿ − 1 m²

n−1

X

k=0

βk

(logm)^k+1(2−σ)^n−1−k, whereβk= _(n−1−k)!^(n−1)! . This implies that for eachm≥2 we get

Z 2

σ

(u−σ)ⁿ⁻¹

m^u du= (n−1)!

m^σ(logm)ⁿ +On

1 m³²(logm)ⁿ

.

Inserting this in (2.7), and considering that||K||b _∞≤ ||K||1, we obtain the desired result.

3. The Resonator

In this section we will construct the resonator. The construction of our resonator is similar to the resonator developed by Bondarenko and Seip [7, Section 3]. The results presented here are extensions of their results, for a region near the critical line. The resonator is the function of the form|R(t)|², where

R(t) = X

m∈M⁰

r(m)m^−it,

and M⁰ is a suitable finite set of integers. Let σ be a positive real number and N be a positive integer sufficiently large, such that

1

2 ≤σ≤ 1

2+ 1

log logN. (3.1)

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Our resonator will depend onσandN. For simplicity of notation, we write log₂x:= log logxand log₃x:=

log log logx. LetP be the set of prime numberspsuch that elogNlog₂N < p≤exp (log₂N)^1/8

logNlog₂N. (3.2)

We definef(n) to be the multiplicative function supported on the set of square-free numbers such that f(p) :=

(logN)^1−σ(log₂N)^σ (log₃N)^1−σ

1

p^σ(logp−log₂N−log₃N), forp∈P andf(p) = 0 otherwise. For eachk∈

1,· · ·,

(log₂N)^1/8 we define the following sets:

Pk :=

p: prime number such thate^klogNlog₂N < p≤e^k+1logNlog₂N ,

Mk:=

n∈supp(f) :nhas at leastαk := 3(logN)^2−2σ

k²(log₃N)^2−2σ prime divisors inPk

,

M_k⁰ :=

n∈M_k :nonly has prime divisors inP_k . Finally, we define the set

M:= supp(f)\

[(log₂N)^1/8]

[

k=1

Mk.

Note that ifm∈ M andd|m thend∈ M.

Lemma 8. We have that

|M| ≤N,

where|M| represents the cardinality ofM.

Proof. The proof follows the same outline that [5, Lemma 2]. The main difference is the appearance of the term (log₃N)^2σ−1,which is well estimated, whenever (3.1) holds. It allows us to obtain the same estimate for the cardinality ofMas the caseσ= ¹₂. By [5, Eq. (9)-(10)], we have that

[x]

[y]

≤exp y(logx−logy) + 2y+ logx ,

for 1≤y≤xand

2 m

n−1

≤ m

n

,

for 3n−1≤m. By the prime number theorem, the cardinality of eachPk is at moste^k+1logN. Therefore, using the above inequalities and (3.1)

|M| ≤

[(log₂N)^1/8]

Y

k=1 [α_k]

X

j=0

e^k+1logN j

≤

[(log₂N)^1/8]

Y

k=1

2

e^k+1logN [α_k]

≤exp

[(log₂N)^1/8]

X

k=1

3(logN)^2−2σ (log₃N)^2−2σ

1

k+3 + 2 logk

k² +(2σ−1) log₂N

k² +(2−2σ) log₄N k²

!

+ 3k+ log₂N

!

≤exp 3

4 +o(1)

(logN)^2−2σ(log₃N)^2σ−1

!

≤exp 3

4 +o(1)

(logN)(log₃N)^2/^log²^N

! .

Then, forN sufficiently large we get that|M| ≤N.

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Lemma 9. For allk= 1,· · ·,[(log₂N)^1/8]we have, asN → ∞ X

p∈P_k

1

p^2σ = (1 +o(1))

Z e^k+1logNlog₂N

e^klogNlog₂N

1 y^2σlogydy, whereo(1) is independent ofk. In particular, we have that

(d+o(1)) 1

(log₂N)^2σ < X

p∈Pk

1

p^2σ <(2 +o(1)) 1

(log₂N)^2σ, (3.3)

for some constant 0< d <1.

Proof. Using [20, Theorem 13.1], under the Riemann hypothesis we have π(x) =

Z x

2

1

logydy+O x¹²logx ,

whereπ(x) is the function that counts the prime numbers not exceedingx. Then, using integration by parts we get

X

p∈P_k

1 p^2σ =

Z e^k+1logNlog₂N

e^klogNlog₂N

1

y^2σlogydy+O

Z e^k+1logNlog₂N

e^klogNlog₂N

logy y^2σ+¹²dy

!

=

1 +O 1

(logN)^1/4

Z e^k+1logNlog₂N

e^klogNlog₂N

1 y^2σlogydy.

Now we can see that Z e^k+1logNlog₂N

e^klogNlog₂N

1

y^2σlogydy≤ e^klogNlog₂N(e−1)

(e^klogNlog₂N)^2σlog e^klogNlog₂N< 2 (log₂N)^2σ.

On the other hand, we know that (e^klogN)^2σ−1<(logN)^4σ−2≤e⁴for all 1≤k≤[(log₂N)^1/8]. Therefore Z e^k+1logNlog₂N

e^klogNlog₂N

1

y^2σlogydy≥ e^klogNlog₂N(e−1)

(e^k+1logNlog₂N)^2σlog e^k+1logNlog₂N> d (log₂N)^2σ,

for some constant 0< d <1.

The following lemma can be considered as an extension of [7, Lemma 4] to the region (3.1).

Lemma 10. We have 1 X

l∈N

f(l)² X

n∈M

f(n)²X

p|n

1

f(p)p^σ ≥c(logN)^1−σ(log₃N)^σ (log₂N)^σ , for some universal constant c >0.

Proof. The proof is similar to [7, Lemma 4]. For eachk∈ 1,· · ·,

(log₂N)^1/8 we define the following sets:

Lk:=

n∈supp(f) :nhas at mostβk := d(logN)^2−2σ

12k²(log₃N)^2−2σ prime divisors inPk

,

wheredis the constant mentioned in Lemma 9, and L⁰_k :=

n∈L_k:nonly has prime divisors inP_k .

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Finaly, we define the set

L:=M\

[(log₂N)^1/8]

[

k=1

Lk.

Now to prove the lemma, it is enough to show that 1

X

l∈N

f(l)² X

n /∈L

f(n)²=o(1), N → ∞. (3.4)

Indeed, using (3.4) and the fact thatL ⊂ M we get 1

X

l∈N

f(l)² X

n∈M

f(n)² X

p|n

1

f(p)p^σ ≥ 1 X

l∈N

f(l)² X

n∈M

f(n)²min

n∈L

X

p|n

1 f(p)p^σ

≥ 1−o(1) minn∈L

X

p|n

1 f(p)p^σ

= 1−o(1)

[(log₂N)^1/8]

X

k=1

d(logN)^2−2σ 12k²(log₃N)^2−2σ min

p∈P_k

1 f(p)p^σ

≥ 1−o(1)

[(log₂N)^1/8]

X

k=1

d(logN)^2−2σ 12k²(log₃N)^2−2σ

k(log₃N)^1−σ (logN)^1−σ(log₂N)^σ

≥c(logN)^1−σ(log₃N)^σ (log₂N)^σ , for some constantc >0. Therefore, it remains to prove (3.4). Since

L:= supp(f)\

[(log₂N)^1/8]

[

k=1

Mk∪Lk

,

it is enough to prove that whenN → ∞ 1 X

l∈N

f(l)²

[(log₂N)^1/8]

X

k=1

X

n∈Lk

f(n)²=o(1), (3.5)

and

1 X

l∈N

f(l)²

[(log₂N)^1/8]

X

k=1

X

n∈Mk

f(n)²=o(1). (3.6)

First we will prove (3.5). For eachk∈ 1,· · ·,

(log₂N)^1/8 and for any 0< b <1 we have 1

X

l∈N

f(l)² X

n∈Lk

f(n)²= 1 Y

p∈P_k

(1 +f(p)²) X

n∈L⁰_k

f(n)²≤b^−β^k Y

p∈Pk

1 +bf(p)² (1 +f(p)²)

≤b^−β^kexp (b−1) X

p∈Pk

f(p)² 1 +f(p)²

! .

(3.7)

10

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Sincef(p)≤1, using the left-hand side inequality of (3.3) we get X

p∈P_k

f(p)² 1 +f(p)² ≥1

2 X

p∈P_k

f(p)²=

(logN)^2−2σ(log₂N)^2σ 2(log₃N)^2−2σ

X

p∈P_k

1

p^2σ(logp−log₂N−log₃N)²

≥

(logN)^2−2σ 8k²(log₃N)^2−2σ

(d+o(1)).

This implies in (3.7) that 1 X

l∈N

f(l)² X

n∈Lk

f(n)²≤exp d

8(b−1)− d

12logb+o(1)

(logN)^2−2σ k²(log₃N)^2−2σ

! .

Therefore, choosingb close to 1 we obtain 3(b−1)−2 logb <0 and summing overkwe obtain (3.5). The proof of (3.6) is similar. For eachk∈

1,· · ·,

(log₂N)^1/8 and for anyb >1 we get 1

X

l∈N

f(l)² X

n∈M_k

f(n)²≤b^−α^kexp

(b−1) X

p∈P_k

f(p)²

. (3.8)

Using the right-hand side inequality of (3.3) we have X

p∈Pk

f(p)²≤

(logN)^2−2σ k²(log₃N)^2−2σ

(2 +o(1)).

This implies in (3.8) that 1 X

l∈N

f(l)² X

n∈Lk

f(n)²≤exp 2(b−1)−3 logb+o(1) (logN)^2−2σ k²(log₃N)^2−2σ

! .

Finally, choosingb close to 1 we obtain 2(b−1)−3 logb <0 and summing overkwe obtain (3.6).

3.1. Construction of the resonator. Let 0 ≤ β < 1 be a fixed number and consider the positive real number κ= (1−β)/2. Note thatκ+β <1. Letσbe a positive real number andT sufficiently large such

that 1

2 ≤σ≤ 1

2 + 1

log logT.

Then we writeN = [T^κ]. Note thatσ andN satisfy the relation (3.1). Now, letJ be the set of integersj such that

h

1 +T⁻¹^j

, 1 +T⁻¹^j+1 \

M 6=∅, and we definemj to be the minimum of

(1 +T⁻¹)^j,(1 +T⁻¹)^j+1

∩ Mforj in J. Consider the set M⁰ :={mj :j∈ J }

and finally we define

r(mj) := X

n∈M,(1+T⁻¹)^j−1≤n≤(1+T⁻¹)^j+2

f(n)²

!¹₂ ,

for everym_j ∈ M⁰. This defines our Dirichlet polynomial R(t) = X

m∈M⁰

r(m)m^−it.

11

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Proposition 11. We have the following properties:

(i) |M⁰| ≤ |M| ≤N. (ii) X

m∈M⁰

r(m)²≤4 X

l∈M

f(l)². (iii) |R(t)|²≤R(0)²T^κ X

l∈M

f(l)².

Proof. (i) and (ii) follow by the definition ofM,M⁰ and Lemma 8. The left-hand side inequality of (iii) is obvious. The right-hand side inequality of (iii) follows by (i), (ii) and the Cauchy-Schwarz inequality.

3.2. Estimates with the resonator. The proofs of the following results are similar to the case σ = ¹₂. According to the notation in [7] we write Φ(t) =e^−t²^/2. ThenΦ(t) =b √

2πΦ(2πt).

Lemma 12. We have

Z ∞

−∞

|R(t)|²Φ t

T

dtT X

l∈M

f(l)².

Proof. The proof is similar to [7, Lemma 5] and we omit the details.

Lemma 13. There exists a positive constant c >0 such that if G(t) :=

∞

X

m=2

Λ(m)am

m^σ+itlogm is absolutely convergent anda_m≥0 for every m≥2, then

Z ∞

−∞

G(t)|R(t)|²Φ t

T

dt≥c T (logT)^1−σ(log₃T)^σ (log₂T)^σ

minp∈Pap

X

l∈M

f(l)².

Proof. The proof follows the same outline of [7, Lemma 7], replacing [7, Lemma 4] by Lemma 10. We omit

the details.

4. Proof of Theorem 2

Assume the Riemann hypothesis. We consider the parameters defined in subsection 3.1.

4.1. The casen≡1 mod 2. We consider the entire function

K_n(z) = (−1)ⁿ⁻¹² log₂TΦ(2πlog₂T z) which has Fourier transform

Kcn(ξ) =(−1)ⁿ⁻¹²

√2π Φ ξ

log₂T

1. (4.1)

Firstly we need to estimate the following integral Z ∞

−∞

Z ∞

−∞

Sn(σ, t+u)Kn(u)du

|R(t)|²Φ t

T

dt. (4.2)

This follows by the same computations as in [7, Section 5]. We will divide (4.2) into 3 integrals.

1. First integral: Using (2.1), (2.5) and Fubini’s theorem we get Z T^β

−T^β

Z ∞

−∞

|Sn(σ, t+u)Kn(u)|dudt

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= Z T^β

−T^β

Z

|u|≤T^β

|Sn(σ, t+u)Kn(u)|dudt+ Z T^β

−T^β

Z

|u|>T^β

|Sn(σ, t+u)Kn(u)|dudt

n

Z T^β

−T^β

Z 2T^β

−2T^β

|Sn(σ, u)Kn(u−t)|dudt+ Z T^β

−T^β

Z

|u|>T^β

log(2|u|+ 2)|Kn(u)|dudt _n

Z 2T^β

−2T^β

|S_n(σ, u)|du+T^β_nT^βlogT.

Hence, by Proposition 11 we obtain Z T^β

−T^β

Z ∞

−∞

|Sn(σ, t+u)K_n(u)|du

|R(t)|²Φ t

T

dtnT^βlogT R(0)²nT^β+κlogT X

l∈M

f(l)². (4.3) 2. Second integral: Using the fast decay of Φ(t), (2.1) and Proposition 11, it follows that

Z

|t|>TlogT

Z ∞

−∞

|Sn(σ, t+u)Kn(u)|du

|R(t)|²Φ t

T

dt T^κe⁻^(log⁴^T⁾²

Z

|t|>TlogT

Z ∞

−∞

|Sn(σ, t+u)K_n(u)|duΦ t

2T

dt

! X

l∈M

f(l)²

=o(1)X

l∈M

f(l)².

(4.4)

3. Third integral:

Z

T^β≤|t|≤TlogT

Z ∞

−∞

S_n(σ, t+u)K_n(u)du

|R(t)|²Φ t

T

dt

= Z

T^β≤|t|≤TlogT

Z

T β

2 ≤|t+u|≤2TlogT

|R(t)|²Φ t

T

dt +

Z

T^β≤|t|≤TlogT

Z

{|u+t|<^{T β}₂ }∪{|u+t|>2TlogT}

|R(t)|²Φ t

T

dt.

(4.5)

Now using (2.1) and Lemma 12, the last integral can be bounded by Z

T^β≤|t|≤TlogT

Z

{|u+t|<^{T β}₂ }∪{|u+t|>2TlogT}

|Sn(σ, t+u)Kn(u)|du|R(t)|²Φ t

T

dt

Z

T^β≤|t|≤TlogT

Z

{|u|<^{T β}₂ }∪{|u|>2TlogT}

|Sn(σ, u)Kn(u−t)|du|R(t)|²Φ t

T

dt

≤ Z

T^β≤|t|≤TlogT

Z

{|u|<^{T β}₂ }∪{|u|>2TlogT}

Sn(σ, u)Kn

u 2

du|R(t)|²Φ t

T

dt n

Z

T^β≤|t|≤TlogT

|R(t)|²Φ t

T

dtT X

l∈M

f(l)².

(4.6)

Inserting (4.6) in (4.5) we obtain that Z

T^β≤|t|≤TlogT

Z ∞

−∞

Sn(σ, t+u)Kn(u)du

|R(t)|²Φ t

T

dt

= Z

T^β≤|t|≤TlogT

Z

T β

2 ≤|t+u|≤2TlogT

Sn(σ, t+u)Kn(u)du

!

|R(t)|²Φ t

T

dt+On(T)X

l∈M

f(l)². (4.7)

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Therefore, combining (4.3), (4.4) and (4.7) we have that the integral in (4.2) can be written as Z ∞

−∞

Z ∞

−∞

|R(t)|²Φ t

T

dt+O_n(T)X

l∈M

f(l)²

= Z

T^β≤|t|≤TlogT

Z

T β

2 ≤|t+u|≤2TlogT

Sn(σ, t+u)Kn(u)du

!

|R(t)|²Φ t

T

dt.

(4.8)

Now we consider two subcases:

4.1.1. The subcase n≡1 mod 4. . In this case note thatKn(u)≥0 for allu∈R. Then by Lemma 12 and the fact thatSn(σ, t) is an even function we obtain in (4.8)

Z ∞

−∞

Z ∞

−∞

Sn(σ, t+u)Kn(u)du

|R(t)|²Φ t

T

dt+On(T)X

l∈M

f(l)²

≤b T max

T β

2 ≤t≤2TlogT

S_n(σ, t)

! X

l∈M

f(l)²,

(4.9)

for some constantb >0. We define Gn(t) =

∞

X

m=2

Λ(m)

π m^σ+it(logm)ⁿ⁺¹Kcn

logm 2π

. (4.10)

By Proposition 7 and (4.1) observe that Z ∞

−∞

S_n(σ, t+u)K_n(u)du= ReG_n(t) +O_n V1

2(t) + 1 ,

fort6= 0. Therefore, the integral on the left-hand side of (4.9) takes the form Z ∞

−∞

Z ∞

−∞

|R(t)|²Φ t

T

dt

= Re Z ∞

−∞

Gn(t)|R(t)|²Φ t

T

dt+On

Z ∞

−∞

V1

2(t) + 1

|R(t)|²Φ t

T

dt

.

(4.11)

Using Proposition 11, Lemma 12 and the definition ofV1

2(t) we get Z ∞

−∞

V1

2(t) + 1

|R(t)|²Φ t

T

dtT X

l∈M

f(l)². (4.12)

Therefore using (4.11) and (4.12) we have

b T max

T β

2 ≤t≤2TlogT

Sn(σ, t)

! X

l∈M

f(l)²≥Re Z ∞

−∞

Gn(t)|R(t)|²Φ t

T

dt+On(T)X

l∈M

f(l)². (4.13) Now using Lemma 13 (note thatKc_n(t) is a positive real function) with

am=Kcn

logm 2π

1 π(logm)ⁿ, for allm≥2 we obtain that

Re Z ∞

−∞

G_n(t)|R(t)|²Φ t

T

dt≥c T(logT)^1−σ(log₃T)^σ (log₂T)^σ

min

p∈PKc_n logp

2π 1

(logp)ⁿ

X

l∈M

f(l)², (4.14)

14

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for some constantc >0. Note that (3.2) and (4.1) imply min

elogNlog₂N <p≤exp (log₂N)^1/8

logNlog₂N

Kc_n logp

2π 1

(logp)ⁿ 1 (log₂T)ⁿ. Inserting this in (4.14), we obtain in (4.13) that (after simplification)

max

T β

2 ≤t≤2TlogT

Sn(σ, t)≥cn

(logT)^1−σ(log₃T)^σ

(log₂T)^σ+n +On(1),

for some constantcn>0. After a trivial adjustment, changingT toT /2 logT and makingβ slightly smaller, we obtain the restrictionT^β≤t≤T.

4.1.2. The subcase n≡3 mod 4. In this case note thatKn(u)≤0 for allu∈R. Similar to (4.9), using the fact thatSn(t) is an even function we find that

Z ∞

−∞

Z ∞

−∞

Sn(σ, t+u)Kn(u)du

|R(t)|²Φ t

T

dt+On(T)X

l∈M

f(l)²

≤b T max

T β

2 ≤t≤2TlogT

|S_n(σ, t)|

! X

l∈M

f(l)²,

(4.15)

for some constantb >0. Using the functionGn defined in (4.10), by Proposition 7 and (4.1) we get Z ∞

−∞

S_n(σ, t+u)K_n(u)du=−ReG_n(t) +O_n V1

2(t) + 1 .

A similar analysis as in the previous case shows that, by Lemma 13 (note that −Kc_n(t) is a positive real function)

Re Z ∞

−∞

−Gn(t)|R(t)|²Φ t

T

dt≥c T(logT)^1−σ(log₃T)^σ (log₂T)^σ

minp∈P−Kcn

logp 2π

1 (logp)ⁿ

X

l∈M

f(l)², (4.16) for some constantc >0. By (3.2) and (4.1) we have

min

logNlog₂N

−Kcn

logp 2π

1

(logp)ⁿ 1 (log₂T)ⁿ. Inserting this in (4.16) we obtain in (4.15) that (after simplification)

max

T β

2 ≤t≤2TlogT

|S_n(σ, t)| ≥c_n (logT)^1−σ(log₃T)^σ

(log₂T)^σ+n +O_n(1),

for some constantcn>0. After the same trivial adjustment of T andβ as in the preceding case we obtain the desired result.

4.2. The casen≡0 mod 2. We consider the entire function

K_n(z) = (−1)ⁿ²⁺¹(log₂T)²zΦ(2πlog₂T z) which has Fourier transform

Kcn(ξ) = (−1)ⁿ² i (2π)³²(log₂T)ξΦ

ξ log₂T

1. (4.17)

15

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The analysis in this case is similar to the casen≡3 mod 4. Using the fact thatSn(t) is an odd function we obtain that (4.15) holds. Using the functionGn defined in (4.10), by Proposition 7 and (4.17) note that

Z ∞

−∞

S_n(σ, t+u)K_n(u)du= (−1)ⁿ²ImG_n(t) +O_n V1

2(t) + 1 .

This implies that in (4.15) we obtain

b T max

T β

2 ≤t≤2TlogT

Sn(σ, t)

! X

l∈M

f(l)²≥Re Z ∞

−∞

(−1)ⁿ²⁺¹i Gn(t)|R(t)|²Φ t

T

dt+On(T)X

l∈M

f(l)², for some constant b >0. Now, using Lemma 13 (note that i(−1)ⁿ²⁺¹Kc_n(t) is a positive real function for t≥0) it follows that

T max

T β

2 ≤t≤2TlogT

Sn(σ, t)

! X

l∈M

f(l)²

≥c T(logT)^1−σ(log₃T)^σ (log₂T)^σ

minp∈PIm

(−1)ⁿ²Kcn

logp 2π

1 (logp)ⁿ

X

l∈M

f(l)²,

(4.18)

for some constantc >0. By (3.2) and (4.17) we have min

logNlog₂N

Im

(−1)ⁿ²Kcn

logp 2π

1 (logp)ⁿ

1

(log₂T)ⁿ.

Inserting this in (4.18) and doing the same procedure as in the previous cases we obtain the desired result.

Acknowledgements

I would like to thank Emanuel Carneiro for all the motivation and insightful conversations on this subject, to Andriy Bondarenko and Kristian Seip for the comments, and to the referee for a very careful review. The author also acknowledges support from FAPERJ-Brazil.

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Email address:[email protected]

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